Three Results on Mixing Shapes

Let α be a Zd–action (d ≥ 2) by automorphisms of a compact metric abelian group. For any non–linear shape I ⊂ Zd, there is an α with the property that I is a minimal mixing shape for α. The only implications of the form “I is a mixing shape for α =⇒ J is a mixing shape for α” are trivial ones for which I contains a translate of J . If all shapes are mixing for α, then α is mixing of all orders. In contrast to the algebraic case, if β is a Zd–action by measure–preserving transformations, then all shapes mixing for β does not preclude rigidity. Finally, we show that mixing of all orders in cones – a property that coincides with mixing of all orders for Z–actions – holds for algebraic mixing Z2–actions.